Can you solve 4 × ∛125 without a calculator? First find the cube root of 125, then multiply the result by 4. Since ∛125 = 5, the final answer is 20.
This problem combines a cube root with multiplication. The fastest reliable method is to simplify the cube root first and then complete the multiplication. The video also shows a memorable digit-sum clue that points to the number 5.
The Quick Answer: 4 × ∛125 = 20
Break the problem into two easy steps:
Step 1: Find ∛125
A cube root asks: What number multiplied by itself three times equals 125?
Because 5³ = 125:
Step 2: Multiply by 4
Replace ∛125 with 5 and multiply:
Therefore, the final answer is:
The Video’s Quick Digit-Sum Clue
The video uses the digits in 125 to create a quick pattern:
Then subtract the cube-root index, 3:
This gives the correct cube-root candidate. Verify it using multiplication:
After the verification, multiply by 4:
The digit-sum calculation is a memorable clue shown in the video, but the exact reason ∛125 = 5 is that 5³ equals 125.
Why Cube Roots Are Different from Square Roots
A square root asks which number is multiplied by itself two times. A cube root asks which number is multiplied by itself three times.
| Operation | Question | Example |
|---|---|---|
| Square root | What number times itself equals the number? | √25 = 5 because 5 × 5 = 25 |
| Cube root | What number times itself three times equals the number? | ∛125 = 5 because 5 × 5 × 5 = 125 |
The small 3 in the cube-root symbol tells you to look for a third power. When no index is shown, the radical is usually a square root; for a cube root, the index 3 is displayed.
Use Nearby Perfect Cubes
If you do not immediately remember 5³ = 125, compare 125 with nearby perfect cubes:
| Number | Cube | Comparison with 125 |
|---|---|---|
| 4 | 4³ = 64 | Too small |
| 5 | 5³ = 125 | Exact match |
| 6 | 6³ = 216 | Too large |
The exact match is 5³ = 125, so ∛125 = 5.
More Examples with the Same Method
Simplify the cube root first, then multiply by the number outside the radical:
Common Mistakes to Avoid
Do not multiply 4 by 125 and then take the cube root. The expression is 4 × ∛125, so simplify the cube root first:
Do not confuse a cube root with a square root. ∛125 asks for a number that is multiplied three times, while √125 asks for a number multiplied two times.
Finally, use the digit-sum calculation as a quick clue, not as a universal cube-root law. The exact verification is 5 × 5 × 5 = 125.
Practice Problems
Simplify the cube root first, then complete the multiplication:
- 2 × ∛64
- 3 × ∛125
- 5 × ∛216
- 4 × ∛512
- 7 × ∛729
Answer Key
- 2 × ∛64 = 2 × 4 = 8
- 3 × ∛125 = 3 × 5 = 15
- 5 × ∛216 = 5 × 6 = 30
- 4 × ∛512 = 4 × 8 = 32
- 7 × ∛729 = 7 × 9 = 63
Final Takeaway
For a problem such as 4 × ∛125, simplify the cube root before multiplying:
The video’s visual clue is:
Then verify the cube root with 5³ = 125. This combination of a fast pattern and exact verification makes cube-root multiplication easier for homework, quizzes, and mental-math practice.
Frequently Asked Questions
What is 4 × ∛125?
4 × ∛125 equals 20, because ∛125 = 5 and 4 × 5 = 20.
How do you find ∛125?
Find the number that is multiplied by itself three times to make 125. Since 5 × 5 × 5 = 125, ∛125 = 5.
What digit-sum trick does the video show?
The video adds 1 + 2 + 5 = 8 and subtracts 3, the cube-root index, to get 5. This is a quick clue for this example and should be verified with 5³ = 125.
Can the cube root be simplified before multiplication?
Yes. When the number under the cube-root symbol is a perfect cube, simplify it first. For example, 4 × ∛125 becomes 4 × 5 = 20.
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